RELATED APPLICATIONS

This application claims the benefit of the priority of U.S. Provisional Application No. 63/072,786, filed August 31, 2020, which is incorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates to a system and method for cyclically controlling system entropy, enabling nonequilibrium mass and energy transfer and producing useful work. The system and method can be used in isothermal cycles.

BACKGROUND

Ergodicity and chaoticity are fundamental concepts underpinning the classical statistical mechanics. Ergodic theory is the theory of the long-term statistical behavior of dynamical systems that arose out of an attempt to understand the long-term statistical behavior of dynamical systems such as the motions of a group of billiard balls or the motions of the earth’s atmosphere. It is well known that an ergodic and chaotic system can reach thermodynamic equilibrium. For example, when the pressures of ideal gas in two connected containers are the same, entropy is maximized. In contrast, in a nonergodic or nonchaotic system, the steady-state can be inherently nonequilibrium.

Nonequilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be adequately described in terms of nonequilibrium state variables that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Nonequilibrium thermodynamics is concerned with transport processes and with the rates of chemical reactions as well as the coupled processes in physical, chemical, electrochemical, and biological systems. Nonequilibrium thermodynamics has been successfully applied to describe biological and/or chemical processes including protein folding/unfolding, mixture modeling, transport through membranes, polymer solution flows, etc. Ideas from nonequilibrium thermodynamics and the informatic theory of entropy have also been extended to describe general economic systems. It has a wide range of potential applications including clean power generation and clean temperature control (e.g., cooling and heating) with little carbon emission, green transportation without using conventional fuels, and effective energy storage with excellent capacity and safety. It holds great promise in broad applications to energy security, energy efficiency, and sustainability.

BRIEF SUMMARY

The present invention includes one or more Spontaneously Non-Equilibrium Elements, or “SNEE”s, or “nonequilibrium elements”. A SNEE is a structure that can spontaneously cause a system steady-state that is different from the thermodynamic equilibrium state. With appropriate a setting, it can render the cross-influence of thermodynamic driving forces asymmetric. The nonequilibrium element can have a small dimension comparable to or less than the mean free path of particles; have a confining environment; work with charged particles; undergo asymmetric motion, leading to a non-equilibrium steady-state; contain small-sized one-sidedly-swinging gates; change a non-thermal thermodynamic driving force without changing temperature; reduce entropy in an isolated system; comprise nano- structured materials. The device, material, or system can comprise one or more heat-exchange mechanisms and the heat- exchange mechanism can absorb heat from a heat reservoir, or release heat to a heat reservoir.

SNEE-based systems and devices may be used as heat engines. SNEE-based systems can be used for transport, storage, management, and conversion of thermal energy or mass. The energy conversion process can be used for cooling or heating. As heat is converted to other forms of energy or transported, the local temperature tends to vary. The generated useful work or the produced thermal energy can be used as power, be used for heating or cooling, be stored, or be transported. The associated mass transfer can be utilized for transport of matters. Multiple mechanisms and multiple units can work in a system.

In one aspect of the invention, a structure comprising one or more nonequilibrium elements is configured to spontaneously induce within a dynamic system comprising a plurality of particles a system steady state that is different from a thermodynamic equilibrium state, wherein a cross-influence of thermodynamic driving forces on the plurality of particles is rendered asymmetric. The one or more nonequilibrium elements may be configured to spontaneously cause an ordered movement or nonequilibrium distribution of particles within the plurality of particles from thermal motion. The one or more nonequilibrium elements may be configured to further impose constraints on a probability of global microstates or render a probability of local microstates asymmetric in forward and reverse processes. The one or more nonequilibrium elements may cause entropy to converge toward a maximum possible value of steady state in an isolated system or may lead to production of useful work in an isothermal cycle. The one or more nonequilibrium elements may be configured to perform one or more of converting heat to other forms of energy, converting other forms of energy to heat, transporting heat, transporting other forms of energy, and transporting mass. The one or more nonequilibrium elements may be configured for heating or cooling, and may be configured as heat-exchange mechanisms, where the heat-exchange mechanisms are configured to absorb heat from a heat reservoir or release heat to a heat reservoir. In some embodiments, the heat-exchange mechanisms may work with a single heat reservoir.

In some embodiments, the one or more nonequilibrium elements comprise nano- structured materials or devices. The nano- structured materials or devices may have dimensions equal to or less than the mean free path of functional particles within the plurality of particles.

In some embodiments, the one or more nonequilibrium elements may be a confining environment. The confining environments may include one or more of voids, vacancies, pores, tubes, channels, or gaps. The one or more nonequilibrium elements may be disposed at a boundary of one or more large areas.

In some embodiments, the one or more nonequilibrium elements may be configured to be switched on and off, while in other embodiments, the nonequilibrium elements may be configured for cyclic operation.

In some embodiments, the one or more nonequilibrium elements may be configured to be barriers or boundaries of one or more fields. The barrier or boundary configuration may be nonuniform.

The one or more nonequilibrium elements may be configured for asymmetric movement and may comprise one-sidedly-swinging gates. The one or more nonequilibrium elements may be comprise organic chains, or may be a height difference, which may be variable. The one or more nonequilibrium elements may have an asymmetric structure, where at least one side of the asymmetric structure is locally nonchaotic or nonergodic. The one or more nonequilibrium elements may comprise electrically conductive or semi conductive materials with charge carriers. The one or more nonequilibrium elements may be configured to change a non-thermal thermodynamic driving force without temperature variation.

The one or more nonequilibrium elements may be a gas or plasma phase, or a condensed matter. The one or more nonequilibrium elements may be configured to work in a potential field, a gravitational field, or an electric field.

In another aspect of the invention, a nonequilibrium element may be configured to spontaneously induce within a dynamic system comprising a plurality of particles a system steady-state that is different from a thermodynamic equilibrium state, where the nonequilibrium element comprises a nonergodic or nonchaotic barrier that imposes asymmetric driving forces on the plurality of particles. The plurality of particles may comprise liquid or gas molecules within a chamber and the barrier may be an asymmetric membrane.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a system containing a spontaneously non-equilibrium element (SNEE). The particle distribution is intrinsically in a non-Boltzmann form, which spontaneously causes δ ≠ δ ₀ , where δ is the particle distribution ratio and δ ₀ is the Boltzmann factor.

FIG. 2A: In a vertical plane in a gravitational field (g), at steady-state the particles do not follow the Maxwell-Boltzmann distribution if the system is non-ergodic or non-chaotic. FIG. 2B: The Billiard-type model system, wherein elastic particles randomly move in the horizontal dimension in between a higher “plateau” and a lower “plain”, across a small transition step. If the plateau height is much less than the mean free path of the particles (λ _F ), the particle motion in the transition step is non- chaotic and the steady-state particle density ratio does not equal to the Boltzmann factor ( δ ₀ ).

FIG. 3 illustrates Monte Carlo simulation of elastic particles freely moving on the surrounding “plain” and the central “plateau”, across the transition step . The dashed circles indicate the boundaries of the transition step with the outer plain and the central plateau.

FIGs. 4A-4C provide typical time profiles of particle density ratio , with the initial being (FIG. 4A) 0.60, (FIG. 4B) 1.00, and (FIG. 4C) 1.40. FIG. 5 shows typical time profiles of the root mean square velocity of incident and reflected particles at the system boundary.

FIG. 6 shows the steady-state particle density ratio as a function of the /λ _F ratio.

FIG. 7: In an isothermal cycle, the system changes from State I A _P /A _G ] = [0.250, 0.888]) to State II ([0.500, 0.888]), III ([0.500, 1.764]), IV ([0.250, 1.764]), and back to State I.

FIG. 8 shows a typical time profile of the average particle velocity in a reference system with = 0. The parameter setting is: N = 800; m = 1.66 × 10 ^-27 ; d = 1.6 × 10 ^-10 ; = 1.38 × 10 ^-20 ; the system area is 4.446 × 10 ^-15 ; the simulation time step is 10 ^-15 .

FIG. 9 illustrates an exemplary cell assembling process. The cell case is 76.2 mm in diameter.

FIG. 10 is a schematic of the liquid replacement process. Details of the testing cell, e.g., the liquid-conductivity measurement probe, are not shown.

FIG. 11 provides a typical liquid conductivity profile during liquid replacement. The electrolyte was CsPiv. Its concentration was changed from 10 mM to 12 mM.

FIGs. 12A-12E provide experimental results of the CsPiv cells: (FIG. 12A) Cell potential with various initial CsPiv concentrations; the solid lines are obtained from the modified electric wire-in-cylinder capacitor (EWCC) model. (FIG. 12B) Variation of c during charging. The experimental data of charge efficiency (A) and the cross-influence factor (Ψ) with the initial CsPiv concentration of 10 mM (FIG. 12C), 12 mM (FIG. 12D), and 14 mM (FIG. 12E).

FIGs. 13A and 13B are, respectively, an exemplary top view and a schematic of the cross section of a polyamide membrane mounted on a compound O-ring.

FIG. 14 is a schematic of an exemplary experimental setup.

FIG. 15 provides photos of the gas pressure measurement system (upper left), the main system body (right), and a polyamide membrane mounted on a compound O - ring (bottom left). The letters in the photo of system body indicate the vacuum valves.

FIG. 16 is a schematic of an exemplary experimental setup. FIGs. 17A-17D provides experimental data of the pressure difference, ΔP = P ₂ — P _1, where P ₁ and P ₂ are the gas pressures measured by sensors 1 and 2, respectively. In (FIGs. 17A, 17B), the surface-grafted side faces container 2; in (FIGs. 17C, 17D), the surface-grafted side faces container 1.

FIG. 18 shows two isothermal cycles that produce useful work by absorbing thermal energy from a single heat reservoir.

FIG. 19 illustrates different states within an exemplary isothermal cycle without mass exchange with the environment.

FIGs. 20A and 20B diagrammatically illustrate an exemplary particle system as a perspective view and a side view, respectively.

FIG. 21 illustrates an exemplary setup of a Monte Carlo simulation.

FIG. 22 plots the particle flux (j) as a function of

FIG. 23 provides time profiles of (A) the x-dimension and (B) the γ-dimension average particle momentums.

FIG. 24 plots the numerical results of pressure across the left-hand side lateral boundary.

FIG. 25 is a schematic of an exemplary asymmetric nanowire for spontaneous production of electricity from thermal energy.

FIGs. 26A and 26B diagrammatically illustrate spontaneous low-temperature to high-temperature heat transfer, induced by molecular-sized one-sidedly-swinging gates, where FIG. 26A shows cyclic operation and FIG. 26B shows a continuous process.

FIG. 27 shows spontaneous low-temperature to high-temperature heat transfer, induced by nanochaotic step in a gravitational field (g).

FIG. 28 shows spontaneous low-temperature to high-temperature heat transfer across a nonchaotic step, when is relatively small.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION Definitions: Unless defined otherwise, all technical and scientific terms used herein have the plain and ordinary meaning as would be understood by a person of skill in the art. Unless expressly limited, where a term is provided in the singular, the description contemplates the plural of that term, i.e., one or more, and where a term is provided in the plural, the inventors also contemplate the singular of that term. To provide a clarifying example, when an object is described, unless that object is expressly described as a single object, “one or more object”, or “at least one object” also falls within the meaning of that term.

A “material” or “device” or “structure” can be in any state or phase of matter, including but not limited to solid phase, liquid phase, gas phase, plasma phase, glass, liquid crystal, condensate, neutron-degenerate matter, quark-gluon plasma, solution, suspension, composites, mixture, particles, fibers, wires, rods, layers, blocks, multilayers, and any component or combination thereof.

“Component” or “device” or “structure” includes, but is not limited to a system, sub-system, parts, sections, solute, material, solvent, container, electrode, solution, membrane, and wire, can comprise metals, alloys, ceramics, carbon, acidic materials, basic materials, salts, polymers, elastomers, composite materials, cations, anions, charged particles, non-charged particles, electrolytes, non-electrolytes, ionic liquids, or any combination or component thereof, unless the context clearly dictates otherwise.

“Heat engine” indicates any system, material, structure, or device that can store, transport, or manage thermal energy, or convert thermal energy to other forms of energy.

“Thermodynamic equilibrium state” indicates the equilibrium state predicted by the conventional statistical mechanics for an ergodic and chaotic system.

“Metal” includes any metallic materials, such as metals, alloys, or a material or device that uses a metal or an alloy.

“Non-equilibrium state” indicates a steady-state that is different from the thermodynamic equilibrium state. “Non-equilibrium element” indicates a component which induces or can induce a non-equilibrium system state.

“Heat reservoir” indicates a material, a device, a system, a structure or an environment that can exchange heat with the material or device, or the heat engine.

“Useful work” or “useful energy” indicates electrical, magnetic, mechanical, potential, chemical, optical, acoustic, or any other non-thermal energy, or any combination thereof. “Useful work” or “useful energy” effects flow or movement within the system, for example, fluid motion, or a transfer or transaction.

“Nonchaotic” is used to describe an area, a material, or a device in which, either globally or locally, no extensive particle collision happens, or the particle collision does not to lead to a random microscopic behavior; a nonchaotic element is also a nonequilibrium element. Discussion on “nonchaoticity” can also be applied to global or local nonergodicity.

“Particles” are things (e.g., objects, items, units, packets, or elements) that exist and/or are acted upon by thermal energy or useful energy within a dynamical system. Particles can be atoms, molecules, clusters of atoms or molecules, charge carriers (e.g., electrons, ions, and holes), subatomic particles, fundamental particles, or any component or combination thereof. To provide a few non-limiting illustrative examples, in a heat transfer study, the particles may be gas molecules acted on by thermal energy. In a billiard system, the particles may be billiard-like balls acted on by, e.g., gravity, lifting force, and centrifugal force. In an economic system, a “particle” can be a good, service, or material that can be the subject of a transaction or transported.

“Structure”, when used with regard to a SNEE, is any physical, mechanical, electrical, chemical, energy -based, or work-based barrier, boundary, or field that induces, or can be used to induce, spontaneous nonequilibrium within a dynamic system. A structure need not be a physical or mechanical object but can be an action or energy that imposes constraints that can induce asymmetry within the system.

“Large area,” when used with regard to a SNEE, can be any one-dimensional, two-dimensional, or three-dimensional zone with a nontrivial size. The boundary of the area can be entirely or partly occupied by the SNEE. SNEE may impose a barrier to all or a part of the particles moving into or out of the area.

FIG. 1 is a diagram representing a nonergodic or nonchaotic barrier. Such a component will be referred to as the spontaneously non-equilibrium element or “SNEE”. A SNEE offers a mechanism to spontaneously achieve a system steady-state that is different from the thermodynamics equilibrium state, without the need for specific knowledge of system microstate.

SNEE-based systems and devices may be used for transport, storage, management, and conversion of thermal energy or mass. The energy conversion process can be used for cooling or heating, for production of useful work, power, or energy, or for transportation of particles. As heat is converted to other forms of energy or transported, the local temperature tends to vary. The generated useful work or the produced thermal energy can be used as power, be used for heating or cooling, be stored, or be transported. The associated mass transfer can be utilized for transport of matters. Multiple mechanisms and multiple units can work in a system. The following examples provide illustrations of the inventive method and its implementation in simulated and practical applications.

Example 1 : Computer simulation of a billiard-type system in gravitational field

FIGs. 2A-2B show a billiard-type SNEE-based system. A large number of elastic particles randomly move in the horizontal dimension. A uniform gravitational field (g) is along the out-of-plane direction, — The central area is higher, which will be referred to as “plateau”. The surrounding lower area will be referred to as “plain”. The plateau height ) can be changed by a lifting force on the plateau, F _G . The total particle number N = N _P + N _G is constant, with subscripts “P” and “G” indicating the surrounding plain and the central plateau, respectively. The plain area (A _P ) can be adjusted by the in-plane pressure (P) at boundary; the area of plateau (A _G ) is fixed. The thermodynamic driving forces under investigation are P and F _G , with the conjugate variables being —A _P and respectively. It is assumed that i) the particle motion is frictionless; ii) the changes of and A _P are reversible; iii) the transition step is smooth; i.e., as particles move across it, no energy is dissipated; and iv) the system boundary is a perfect heat exchanger, and the environment is an infinitely large heat reservoir with a constant average particle kinetic energy . Moreover, v) we choose to study a system where is much smaller than the plain and plateau sizes; when g = 0, has little influence on the particle distribution.

As shown in FIG. 2B, the plateau and the plain are two large areas separated by the transition step and are respectively dominated by F _G and P. F _G is directional and asymmetric in the transition step. Variations in A _P or would cause particle redistribution across the transition step, resulting in the exchange of particle kinetic energy (K) with the environment, A _P and can be adjusted reversibly and independently, with a constant . If is much less than the particle mean free path (λ _F ), the transition step would become a SNEE. In this example, the SNEE imposes an energy barrier. Because the transition step is enclosed in the interior, pressure (P), area change (dA _P ), force (F _G ), and displacement can be readily measured at the system outer boundary.

The in-plane pressure of the plain is governed by P·A _p = N _P · . The lifting force (F _G ) on the plateau contains two components: the particle weight F _Gg = mgN _G , and the centrifugal force (F _Gc ) caused by the particles changing direction in the transition step, where m is the particle mass and g is the gravitational constant. Denote the characteristic time for a particle to change direction in transition step by . At the steady-state, during on average particles pass through the transition step, where represents the average z-dimension velocity component of these particles and L _G is the plateau perimeter. The average centrifugal force per particle is on the scale of Thus, where and D _G is the plateau size. When D _G is much larger than ), F _Gc is negligible compared with F _Gg , since is on the same scale as . Hence, the lifting force is simplified as F _G = mgN _G .

At the steady-state, the second law of thermodynamics requires that , where However, the particle density ratio (/?) is considerably influenced by the ergodicity and chaoticity of the transition step. In the SNEE transition step where is much less than the nominal mean free path of particles (λ _F ), the relevant kinetic energy for particles to overcome the gravitational energy barrier from the plain to the plateau is mostly determined by the -dimension momentum, in average less than the total particle momentum by a factor of 2. Consequently, the effective crossing ratio ( δ) is reduced from the Boltzmann factor ( δ ₀ ) to about In accordance with and N = N _G + N _P , we have and . Thus, in an isothermal cycle wherein P and F _G work alternately, the work produced by P (W _P ) is greater than the work consumed by F _G (W _G ). In the ideal case scenario that

FIG. 3 diagrammatically illustrates an exemplary computer simulation setup. Elastic particles freely move in a square box in the plane. When a particle impacts the system boundary, it will be reflected along a random direction with a random velocity (v); the reflected velocity follows the 2D Maxwell-Boltzmann distribution (p(v)) with a constant The simulation box is divided into two areas by a narrow circular band: the surrounding plain and the central plateau. The circular band is the transition step, in which the local dimension is denoted by along the radius direction toward the center. No force is applied on the particles on the plain and the plateau, except for particle collision. The particles in the transition step are subject to a constant force, mg, along The width of the transition step is For each simulation case, at each time step, the particle numbers on plain ( N _P ) and plateau ( N _G ) are counted. Average is computed for every 10 ⁴ ~4×10 ⁴ time steps; the lifting force (F _G ) is calculated as average mgN _G for every 10 ³ ~2.5×10 ³ time steps; the in-plane pressure (P) is obtained as where “∑” indicates summation of all the particle-boundary collisions during time steps, L ₀ is the length of system boundary, and is the change in particle momentum in the normal direction. Initially, the particles are evenly placed on the plain and the plateau, and the particle velocity follows p(v) and the direction is random. Each simulation case is continued until the steady-state of P, F _G , and has been established. The initial does not affect the steady-state. FIGs. 4A-4C provide typical time profiles of particle density ratio , with the initial being (FIG. 4A) 0.60, (FIG. 4B) 1.00, and (FIG. 4C) 1.40. The steady-state particle density ratios are 0.59, 0.59, and 0.61, respectively. The parameter setting is similar to Case Rl in FIG. 6 (see Table 1 below) except that the gravitational constant (p) is two times smaller. At the steady-state, there is no overall heat exchange with the environment, as shown in FIG. 5 based on the parameter settings for Case R1 in Table 1.

FIG. 6 shows the calculated as a function of the / λ _F ratio. The simulation case numbers (R1 to R7) are shown, with the parameter settings listed in Table 1. For different simulation cases, the nominal mean free path of particles, λ _F is adjusted by changing the particle diameter (d) and the plain area (A _P ). The value of is also varied, and g is controlled to keep = 0.5, so that the Boltzmann factor ( δ ₀ ) is constant 0.607.

Table 1 Parameter settings for the simulation cases in FIG. 6

The total particle number is N = 800; = 1.38 × 10 ^-20 ; m = 1.66 × 10 ^-27 ; the simulation time step is Δt ₀ = 10 ^-15 ; the normalization factor of the particle diameter (d) is d ₀ = 1.6 × 10 ^-10 ; the plateau area is A _G = 1.256 × 10 ^-15 , which is used to normalize the plain area (A _P ); the normalization factor of = 10 ^-9 ; the normalization factor of the gravitational constant (p) is g ₀ = 4.16 × 10 ¹⁵ . The units of these parameters may be taken as the SI unit system. The error bars are calculated as the confidence interval, is the average value, n _t is the number of data points, s _t is the standard deviation, and δ _t is the inverse of student’s t-distribution with the confidence level of 0.9. When is close to = 0.368, suggesting that the particle motion along z is dominated by the -dimension momentum. The trend is clear that increases with /λ _F , especially in the range of /λ _F from 1 to 4. When /λ _F is relatively large, converges to δ ₀ = 0.607. From these results, it can be seen that the ergodicity of the transition step significantly affects is close to the Boltzmann factor /λ _F is relatively large (Case R6 and Case R7), and close to when /λ _F is relatively small (Case R1 and Case R2).

FIGs. 7A and 7B plot an isothermal operation cycle. State I is the same as Case R1 in Table 1 and in FIG. 6, with /λ _F ≈ 0.1. The solid lines are the regression curves. At State I, = [0.25, 0.888] and /λ _F ≈ 0.1. From State I to II, A _P is constant and increases to 0.5. As rises, less particles are on the upper plateau, so that F _G decreases while P becomes larger. From State II to III, is constant and A _P /A _G expands to 1.764. Since the particle density is reduced, both F _G and P are smaller. From State III to IV, A _P does not vary and is lowered back to 0.25 as seen in FIG. 7B. Because the energy barrier of transition step is less, F _G increases and P decreases. Finally, the system returns to State I, and the densification of particles leads to the increase in both F _G and P. Table 2 lists (FIG. 7A), A _P /A _G (FIG. 7B) as well as the steady-state F _G , P, and for each simulated system state. In addition to States I, II, III, and IV, there are 12 intermediate states in between them, marked as I-a, I-b, etc. For all the system states, d = d ₀ and g = g ₀ . The error bars are calculated as the confidence interval,

From State I to II (through States I-a, I-b, and I-c) and from State III to IV (through States III-a, III-b, III-c), the plain-to-plateau area ratio (A _P /A _G ) remains constant. As varies, the particle density ratio changes. The lifting force can be expressed as F _G = mgN _G The simulation data of approximately fit with exp where captures the effect of /λ _F . If = 1, the right-hand side of the equation is reduced to the Boltzmann factor. Because the crossing ratio ( δ) of the SNEE transition step is less than δ ₀ , is generally larger than 1. Hence,

In this equation, is the only adjustable parameter for data fitting; all other parameters are known (Table 2).

Table 2 State evolution of the isothermal cycle in FIG. 7A-7B

When is set to 2.28, it agrees well with the MC simulation data of the F _G — relationship from State I to II; when is set to 2.06, it agrees well with the simulation data from State III to IV, as shown by the upper and lower solid curves in FIG. 7A.

From State II to III (through II a, II b, and II-c) and from State IV to I (through IV-a, IV-b, and IV-c), is kept constant while the plain area (A _P ) changes. The in-plane pressure can be assessed as where is the average particle density ratio and = 9.039 x 10 ^-21 is calculated from a reference system with = 0 (FIG. 8). For the upper solid curve in FIG. 7B, is set to 0.3755, the average of

States II, II-a, II-b, II-c, and III; for the lower solid curve, is set to 0.6100, the average of States IV, IV-a, IV-b, IV-c, and I. This equation is in agreement with the simulation results of the P — A _P relationship.

The F _G — loop consumes work which is computed to be 3.20×10 ^-19 (nearly 23.2 ), as the area enclosed by the upper and lower

F _G — curves in FIG. 7A. The P — A _P loop produces work = 4.79×10 ^-19 (nearly 34.7 , calculated as the area in between the upper and lower P — A _P curves in FIG. 7B. The work produced by P (W _P ) is greater than the work consumed by F _G (W _G ); the ratio between W _P and W _G is W _P /W _G =1.497. After a complete cycle, the overall work production is W _tot = W _P —W _G = 1.59×10 ^-19 (nearly 11.5 . The normalization factors are

Example 2: Large ions confined in small nanopores

In this sample application of the inventive approach, we measured the mismatch between the charge efficiency and the ion-concentration sensitivity of electric potential of large ions adsorbed in charged micropores.

Spectracarb-2225 Type-900 nanoporous carbon was processed into 15.88-mm- diameter electrode discs and dried in a gravity convection oven (VWR, 1330GM) at 120 °C for 24 hours. The disc mass (m _e ) was around 25 mg. Two carbon discs were immersed in 20 mL electrolyte solution in a 20-mL vial in a vacuum oven (VWR, Shel-Lab 1410) at 94.8 kPa for 5 min. The electrolyte was cesium pivalate (CsPiv). Membrane separators (Dreamweaver, Titanium 30) were cut into 17.46-mm-diameter discs and soaked in the same electrolyte solution for 10 min. Two spacer rings were cut from a 415 -μm -thick polycarbonate film (McMaster, 85585K15) by 7.14-mm-inner-diameter and 15.88-mm- outer-diameter punch heads.

Two 25.4-mm-thick 76.2-mm-diameter polyacrylic discs (McMaster, 1221T63) were used as cell cases. Eight equally spaced through-holes were drilled by using a Palmgren drill press (McMaster, 28015A51), with a 7.14-mm-diameter drill bit (McMaster, 2901A126). The center-to-center diagonal distance was 50.8 mm. A center hole was drilled on each cell case by using a 3.18-mm-diameter drill bit (McMaster, 2901 A115) for liquid replacement.

External connection tubes were fabricated by fitting 200-mm-long 0.50-mm- inner-diameter 1.52-mm-outer-diameter ethylene-vinyl acetate (EVA) tubes (McMaster, 1883T1) into 50-mm-long poly(vinyl chloride) (PVC) tubes (McMaster, 5231K124), with epoxy adhesive (J-B Weld, McMaster, 7605 A18) applied at the tube interfaces. After curing at ambient temperature for 10 h, a connection tube was inserted into the center hole of the top/bottom cell case, secured by epoxy adhesive. A 0.6-mm-diameter needle (BD PrecisionGlide, 305194) was tightly pressed into the EVA tubing, with the other end connected to a 1-mL syringe (BD, 309659).

A pyrolytic graphite sheet (Panasonic, EYG-S121803) was cut into 1.5x20 mm strips by a stainless-steel razor blade (McMaster, 3962A48). A 25-μm-thick nickel foil (MTI, MF-NiFoil-25u) was cut into 2×30 mm strips, followed by repeated flattening in a rolling mill (Durston, DRM F 150) with a 20-μm gap. Electrical outlet tab was produced by attaching a graphite strip to a nickel strip using a 4-mm-wide Kapton polyimide tape (McMaster, 7648A32), affixed on the bottom cell case, with the overlapping length of ~10 mm.

A liquid-conductivity measurement probe was fabricated by using two 0.5-mm- wide 50-mm-long nickel strips 1 mm apart from each other, with the gage length of ~10 mm. The strips were cut from the aforementioned nickel foil and repeatedly flattened by the rolling mill through a 20-μm gap. The strips were fixed together in parallel by three 0.8-mm-wide Kapton tapes. The probe was sandwiched in between the two membrane separator discs.

The cell assembling procedure is shown in FIG. 9. Stainless steel socket head screws (McMaster, 92196A821) were fit into the through-holes in the bottom cell case, with 1.62-mm-thick nylon plastic washers (McMaster, 95606A420). Two electrical outlet tabs were affixed onto the edge of the bottom cell case by using Kapton polyimide tapes (McMaster, 7648A32). The positive electrode disc was placed at the center of the cell case, covered by a separator-probe sandwich and a negative electrode disc. The tab was adjusted to ensure an adequate electrical connection. The electrode stack was covered by two layers of spacer rings, to reduce the free space and to enhance the electrical contact. Two layers of nylon plastic shims (~0.785 mm thick, McMaster 90295A450) were added to the screws on the bottom cell case. Before placing the top cell case, the electrode stack had been enclosed by a Viton Fluoroelastomer sealing o- ring (McMaster, 8297T174). Stainless steel nuts (McMaster, 91849A029) were fastened on the screws with 1.62-mm-thick nylon plastic washers (McMaster, 95606A420), by a L-key (McMaster, 5503A22) and a wrench (McMaster, 7152A812), until the shims contacted the top and bottom cell cases. Excess solution was extracted by the syringes via the connection tubes.

After the cell was assembled, the electrical outlet tabs were connected to a Neware CT-ZWJ-4S-T Analyzer. All the cells were pre-cycled between 0 to 0.8 V at 0.1 mA for 20 cycles. Cells with the coulombic efficiency lower than 98% or the internal impedance higher than 50 Ω were rejected. Coulombic efficiency was defined as the ratio of discharging capacity to charging capacity of ε — Q cycle. Internal impedance was calculated from the cell potential change before and after a known current is applied. Table 3 shows typical internal impedance of the CsPiv cells.

Table 3 Typical impedance of the CsPiv cells

The prepared cell was tested in charge-discharge cycles, with a constant charging- discharging rate of 0.1 mA. The cell potential was continuously monitored. Charging was stopped regularly to measure the cell potential (ε). At each stop, the cell was rested for 1 min; ε was defined as the open-circuit voltage at the end of the resting period. To measure the liquid conductivity the embedded liquid-conductivity measurement probe was connected to a Hanna HI5321-01 Electrical Conductivity Meter by copper wires. The conductivity was recorded at the same time as the cell potential.

On each cell, the measurement of ε and was repeated for various initial electrolyte concentrations. The initial CsPiv concentrations were 10 mM, 12 mM, 14 mM, and 16 mM. For each condition, three nominally identical cells were tested. After a charging measurement was completed, the liquid phase was replaced by another solution of the same electrolyte but a slightly different concentration (c). Liquid replacement involves generating a slow constant-concentration flow for a certain period of time, until a new equilibrium is reached. It could precisely adjust the electrolyte concentration, with minimum influence on other cell components, such as the electrode stack and the electrical connections. FIG. 10 diagrammatically depicts the liquid replacement system. The testing cell was first connected to two 60-mL syringes (“A” and “B”). Syringe “A” contained the electrolyte solution to be filled into the cell. It was compressed by an Instron-5582 machine, to generate a flow with a constant rate of 4 mL/min for 15~20 minutes, until the liquid conductivity inside of the cell was stabilized at the new level. The liquid conductivity was continuously monitored by the measurement probe embedded in the cell. Outflow electrolyte solution was collected by syringe “B” on the other side. FIG. 11 shows a typical liquid conductivity curve. After liquid replacement, the cell was cycled again for 20 cycles between 0 to 0.8 V. Cells that failed to meet the criteria of coulombic efficiency (> 98%) and internal impedance (< 50 Ω ) would be abandoned. Similar cell potential and measurement was conducted with the new liquid phase. The measurement was repeated for multiple electrolyte concentrations. Ambient temperature was ~23 °C for all the tests.

The specific electrode charge was obtained as Q = 3.6It _c /m _e , where 3.6 is the factor accounting for unit conversion, I is the charging current in mA, t _c is the charging time in hr, and m _e is the electrode mass in gram. For CsPiv, the electrolyte concentration, c, was related to the measured by using the calibration curve: - 0.0934. The units of c and are mM and mS/cm, respectively. The calibration curves were measured by using the setup shown in FIG. 10. Liquid replacement was carried out for 15~20 min with solution of known electrolyte concentration (c), until has converged. The measurement of was repeated at various c: 1 mM, 2 mM, 4 mM, 6 mM, 8 mM, 10 mM, 20 mM, 30 mM, 40 mM, and 50 mM.

The cell volume was V _cell = -A _c d _c , with A _c = 197.9 mm ² being the cross-sectional area of electrode stack and d _c = 0.74 mm the thickness of cell cavity. The solid volumes of carbon (V _c ) and membrane separators (V _SP ) were calculated from their mass densities: V _c = m _e /ρ _c and V _SP = m _SP /ρ _sp , with ρ _c = 2.2 g/cm ³ being the density of carbon, m _SP the mass of separators, and ρ _sp = 1.6 g/cm ³ the density of ligament material. The liquid volume was taken as V _L = V _cell — V _c - V _SP , typically around 120 μL. Calculation of the charge efficiency followed the established procedure in literature: A = —ξV _L Δc/ΔQ, where ξ is the Faraday constant, Δc is the measured variation in molarity, ΔQ = 3.6/Δt _c /m _e is the change in electrode charge, and Δt _c is the duration associated with Δc.

FIGs. 12A-12E plot the measurement results. The cross-influence factor (Ψ) was calculated as where z = 1 is the ion charge, R is gas constant, T is ambient temperature, Δε is the difference in cell potential at the same charging state of two adjacent ε — Q curves in FIG. 12A, and Q is the electrode charge. The two adjacent ε — Q curves are measured from the same cell. The only difference is that the initial concentration of one curve is higher than that of the other by 2 mM. The difference in concentration, Ac, was calculated from the liquid conductivity change. The value of c is taken as the lower bound of Ac. FIG. 12B plots variation of c during charging. The experimental data of charge efficiency (Λ) and the cross-influence factor (Ψ) with the initial CsPiv concentrations of 10 mM, 12 mM, and 14 mM, as shown in FIGs. 12C- 12E, respectively. It can be seen that Λ ≠ ψ; that is, the cross-influence of the thermodynamic forces (the cell potential ε and the chemical potential μ ) is asymmetric, and the ion distribution is spontaneously non-equilibrium. Specifically, where N _e is the ion amount in the electrolyte solution. That is, the measured sensitivity of ε with c, , is greater than what the second law of thermodynamics dictates. With the large ions and the small nanopores, as c is varied, the change in ε is significantly . The “excess” electric energy is converted from thermal energy from the environment.

Since Λ ≠ ψ, as the ion concentration is reversibly adjusted, the consumed work for the control of μ would be less than the produced work of ε. The operation cycle can begin from Step 1, in which the ion concentration in electrolyte solution is increased by using an osmotic membrane as piston to remove a portion of solvent (water). As c inreases, ε is reduced. Then, in Step 2, the electrodes are charged by a certain charges, ΔQ. As the electrodes adsorb ions, c is reduced. In Step 3, the osmotic-membrane piston is moved back, so that solvent is added in the system and c decreases. In Step 4, the electrodes are discharged, and the system returns to the initial state. The discharged electric work in Step 4 is more than the charged electric work in Step 2; the input mechanical work of osmotic-membrane piston in Step 1 is more than the output mechanical work in Step 3. All of the operations are reversible. The overall generated electric energy is more than the overall consumed mechanical work of osmotic- membrane piston motion.

The difference between Λ and ψ indicates that the distribution of the adsorbed ions in the micropores is nonequilibrium. It breaks the symmetry of the cross-influence of the chemical potential and the electrical potential. The nonequilibrium distribution should be attributed to the confinement effect of the micropore walls, where the ion configuration and movement are in low-dimensional forms. Example 3 : Asymmetric gas permeability of microporous membrane

In this example implementation of the inventive scheme, we measured the effective gas permeability of an asymmetric microporous membrane.

Toray UTC-82V polyamide (PA) microporous membrane was obtained from Sterlitech. A membrane sample was sectioned by a razor blade, about 1.7 cm large. It was firmly attached to the stainless steel inner frame of a McMaster-4518K63 compound o-ring (FIGs. 13A-13B), using McMaster-7541 A77 Devon epoxy. The epoxy was cured at room temperature for 24 h. The membrane was thoroughly cleaned by deionized (DI) water and immersed in 50 wt% aqueous solution of isopropyl alcohol (IPA) for 24 h. Untreated membrane was dried in a JeioTech ON-01E-120 oven at 75 °C for 30 min. For surface treatment, lauric aldehyde (LA) and sulfuric acid (H2SO4) were purchased from Sigma Aldrich (CAS No. 112-54-9 and CAS No. 7664-93-9, respectively). 20 mM aqueous solution of LA was prepared, and H ₂ SO ₄ was dropped in to adjust the pH value to 2. About 1 ml LA solution was added onto the PA membrane surface, filling the steel frame. The setup was heated at 75 °C for 30 min in a JeioTech ON-01E-120 oven. Then, the LA solution was removed and the membrane was repeatedly rinsed by DI water, immersed in DI water at 50 °C for 2 h, dried at 75 °C for 30 min, and rested at ambient temperature for 24 h. The Viton fluoroelastomer outer ring was placed onto the steel inner frame. FIG. 14 and FIG. 15 are schematic and photographic images, respectively showing details of the experimental setup. Table 4 lists the major components.

Table 4 Parts list of the experimental system

“NW-25” indicates that the diameter is 25 mm.

The containers mainly consisted of thin-walled stainless steel vacuum hoses, four-way connectors, and flexible couplings, and were connected to a MTI EQ-FYP- Pump-110 vacuum pump, two Inficon SKY-CDG200D pressure sensors, and a pentafluoroiodoethane ( C ₂ F ₅ I) gas storage vessel (Sigma Aldrich, CAS No. 354-64-3). Vacuum clamps (see the inset at the upper-right comer of FIG. 14) and vacuum grease are used at all the connections. All the connections and valves were carefully adjusted, until satisfactory reference curves were obtained. The compound o-ring with one-sidedly surface-grafted membrane was placed in between valves B1 and B2. An untreated membrane was mounted on a similar compound o-ring and placed in between valves C1 and C2. FIG. 16 depicts the assembled system. In some tests, the untreated membrane between valves C1 and C2 is replaced by a non-permeable solid polycarbonate film. The inset on the right-hand side of FIG. 16 provides a magnified view of dodecyl chains covering micropores. The organic chain tends to be pushed close by gas molecules moving toward the micropore from right to left, while can be pushed open in the inverse direction.

Valve G was closed, and all the other valves were open. The vacuum pump was turned on. The gas pressure was reduced to below 0.06 mTorr and kept for ~1 h. The pressure sensors were calibrated. Valve P was closed, and the pump was turned off. Valve G was opened, and C ₂ F ₅ I gas slowly flew into the containers, until the pressure sensor readings reached ~0.8 Torr. Valve G was closed, and the system rested for 2 h. If we needed to change a membrane, the valves across it would be closed and the gas pressure was maintained in the rest of the system. After the membrane change, the operation of vacuum pump was repeated.

FIGs. 17A-17D plot the testing results. The black curves (labeled “BLACK”) are for one-sidedly surface-grafted membrane; the red curves (labeled “RED”) are for untreated symmetric membrane; the gray curves (labeled “GRAY”) are for non- permeable film. The hashed arrows indicate that valves A1 and A2 are closed; the solid arrows indicate that A1 and A2 are opened.

To measure the BLACK curves in FIGs. 17A, 17C, before the measurement, valves A1, A2, B1, and B2 were open, and other valves remained shut. After P ₁ and P ₂ had been stabilized at ~0.8 Torr, valves A1 and A2 were closed. The operation of the two valves should be slow and as simultaneous as possible, to minimize the disturbance on ΔP. The readings of P ₁ and P ₂ were recorded, until the new steady-state has been reached for ~5 min. Then, valves A1 and A2 were opened again, and the process were repeated for 3 times. The measurement procedures of the RED and the GRAY curves in FIG. 17A were similar, except that valves B1 and B2 remained shut and valves C 1 and C2 remained open. For the gray curve, the membrane in between valves C1 and C2 was replaced by a non-permeable 250-μm-thick solid polycarbonate film (McMaster 85585K103). To measure the curves in FIG. 17B, 17D, the initial pressure difference between the two containers were set to about 2 or -2 mTorr. After P ₁ and P ₂ had been stabilized at the same ~0.8 Torr, valves B1 and B2 remained open and all the other valves were closed. Then, valve P was opened, and the vacuum pump was turned on to reduce the pressure in the hose between valves A1 and A2. After ~5 sec., valve P was closed. Valve A1 or A2 were then opened, so that gas in container 1 or 2 flew into the section between valves A1 and A2, and P ₁ or P ₂ decreased by ~2 mTorr. The continued changes of the pressure sensor readings were monitored.

For each membrane sample, before surface treatment and further testing, its gas permeability was observed. The procedure was similar to the measurement of FIG. 17B, except that valves B1 and B2 remained shut and valves C1 and C2 remained open. The untreated membrane was installed between valves C1 and C2. The initial ΔP was set to ~2 mTorr. If the decrease rate of ΔP was different from 5~10 μTorr/sec by more than 50%, the sample would be rejected. About 1/5 of the membrane sheets met this criterion.

As the initial P ₁ and P ₂ were balanced, valves A1 and A2 were closed, leaving only valves B1 and B2 open. FIG. 17A shows that spontaneously, a pressure difference was built up across the one-sidedly surface-treated membrane. In ~9 min, ΔP reached ~1.2 mTorr. It was positive, indicating that P ₂ > P ₁ , that is, gas flew from the low- pressure side (container 1) to the high-pressure side (container 2), until the steady-state was reached. The effective gas permeability can be defined by j/( P ₁ — P ₂ ), with j being the average gas flow rate; it was negative. After ΔP has stabilized for ~5 min, valves A1 and A2 were opened. The pressure difference was instantaneously reduced to zero. As valves A1 and A2 were closed and opened again, the increase and decrease of ΔP were repeatedly observed. If the same membrane was flipped and the surface-grafted side faced container 1, similar ΔP profiles could be measured, except that ΔP = P ₂ — P ₁ was negative (FIG. 17C).

Because ΔP eventually reached a steady-state and the sign of the steady-state ΔP was dependent on the membrane direction, it must not be caused by the leakage of containers or valves. The final ΔP was unrelated to the initial pressure difference (FIGs. 17B, 17D), suggesting that the steady-state was determined by the crossing ratio of the membrane (A;). The container volume was ~710 cm ³ , compared with which the amount of gas adsorbed by the ~1.3 cm ² membrane specimen was negligible. The pressure sensors were ~90 cm away from the membrane, ensuring that the measured ΔP was not a local phenomenon. If valves C1 and C2 were open and all the other valves were closed, ΔP was around zero, as shown by the RED curve in FIG. 17A. That is, the gas pressure across an untreated symmetric membrane was balanced, as it should be. If we replaced the untreated membrane between valves C1 and C2 by a non-permeable solid polycarbonate film, the change in ΔP was also trivial over time (the GRAY curve) in FIG. 17A, confirming that the two containers were nearly symmetric.

In panel A of FIG. 18, the environment is an infinitely large reservoir of thermal energy and gas molecules, at constant pressure P ₀ and temperature T. The inset at the lower-left corner shows the pressure-volume (P — V) loop; numbers I, II, III, and IV indicate the system states. At State I, the chamber is open to the environment through a venting channel, so that the inner gas pressure is P ₀ . The chamber volume is V _s . From State I to II, the venting channel is closed, and a one-sidedly surface-grafted membrane is exposed; the grafted side faces the inside of chamber. The inner pressure spontaneously rises to κP ₀ . From State II to III, a piston moves out of the chamber and does work to the environment. The chamber volume increases to V _L , while the gas pressure remains κP ₀ . From State III to IV, the membrane is covered, and the venting channel is open, so that the pressure decreases to P ₀ . Finally, the piston moves back, and the system returns to State I. After a complete cycle, the system produces work W _tot = ( κ — 1) P ₀ ΔV, where ΔV = V _L — V _s . The produced work is from the absorbed heat. FIG. 18 panel B depicts another example in which the operation cycle involves two thermodynamic driving forces, the gas pressure (P) and the lifting force ( F _G ). Two chambers are separated by an asymmetric membrane, in a gravitational field (g). The membrane is always exposed. The inset on the right-hand side shows the P — V _P and the F _G — loops. The surface-grafted side is toward the stationary lower chamber. The volume (V _P ) and the gas pressure (P) of the lower chamber change with the piston motion. The upper chamber can be moved along the vertical direction by a lifting force (F _G ). The height of the upper chamber is denoted by which is negligible compared to the chamber size (x _c ). The chamber thickness (z _c ) is much less than . The piston and F _G work alternatingly in cycles, so that V _P varies between varies between z_ and z ₊ . Because z is much less than the chamber size (x _c ), the influence of the volume of the transition step on the distribution of gas molecules is negligible. Since the chamber thickness (z _c ) is much less than z, the gas density in the upper chamber can be calculated as ρ _G = ρ _ρ δ, where p _p is the gas density in the lower chamber, is the density ratio, AT is the membrane crossing ratio, m _g is the gas molecular mass, κ _B is the Boltzmann constant, and T is temperature. In accordance with ρ _G V _G + ρ _P V p = M, we have where M is the total gas mass, V _G is the volume of the upper chamber, and Vp is the volume of the lower chamber. Thus, the lifting force is and according to the ideal gas law, the gas pressure in the lower chamber is From state I to II, F _G lifts the upper chamber from and does work to the system The volume of lower chamber remains constant. From State II to III, the piston expands the lower chamber from and does work to the environment ' The height of the upper chamber remains unchanged. From State III to IV, the upper chamber is lowered back to , and F _G does work to the environment

From State IV back to I, the lower chamber shrinks to Vp and the piston does work to the system W ₄₁ = In each operation cycle of P and F _G , P produces work

, consumes work total produced work is

1, W _tot is positive.

It is apparent that the two systems in FIG. 18 contradict the Kelvin-Planck statement of the second law of thermodynamics, as useful work is cyclically produced from a single reservoir of thermal energy. Panel A of FIG. 18 requires that the nonequilibrium mechanism can be switched on and off. FIG. 18, panel B, needs more than one relevant thermodynamic driving force. There are a number of possible variants in this illustrative example. For example, mass exchange with environment or continued membrane exposure may not be necessary.

Referring to FIG. 19, different states within an exemplary isothermal cycle without mass exchange with the environment are shown. In the pressure-volume (P — V) loop shown in the plot, the upper-case Roman numerals indicate the states of the left chamber, and the lower-case Roman numerals numbers indicate the states of the right chamber. The environment is an infinitely large reservoir of thermal energy at constant pressure (P ₀ ) and temperature (T). There are two identical one-sidedly surface-grafted membranes on the dividing wall between the two chambers. The grafted side of the upper membrane faces the right chamber; the grafted side of the lower membrane faces the left chamber. At State I (upper left), both membranes are covered. The gas pressure in the two chambers is the same P _0; the volumes of the left and the right chambers are respectively and V _o , with κ; being the membrane crossing ratio. At State II (lower left), the upper membrane is exposed. As the gas flows from the left side to the right side, the gas pressures in the left and the right chambers become respectively. Then, both membranes are covered. The left piston moves into the left chamber, and the right piston moves out of the right chamber. The membranes are not exposed during the piston operation. At State III (lower right), the volumes of the left and the right chambers are V0 and respectively. It can be seen that State III is symmetric to State I. The processes from State III to IV and from State IV to I are similar to I to II and II to III, respectively. In each cycle, the total input work is W _in = P ₀ V ₀ lnκ; the total output work is the overall produced work is W _tot =

Another simple system configuration is FIG. 16, with valves A1, A2, B1, and B2 open, and all the other valves being shut. The asymmetric membrane generates a continuous gas flow loop, which can produce useful work by absorbing heat from the environment.

The surface-grafted organic chains are somewhat similar with one-sidedly- swinging gates. The working mechanism of the gate is consistent with the principle of micro-reversibility. For a particle-gate interaction event in which the particle crosses the gate, denote the microstate of incident particle by ψ _a , and the microstate of outgoing particle by ψ _b . When the particle is at ψ _a and ψ _b , the particle velocities are respectively denoted by v _a and v _b , and the microstates of the gate are respectively denoted by <t> _a and O _b . For the reverse process, use to indicate the reverse microstate. Compared with the forward microstate, the velocity direction of the reverse microstate is inverted, with everything else being identical. For each set of ψ _a and Φ _a in the forward process, there are a set of of the reverse process. Time reversibility ensures micro- reversibility: L where {•| ♦} indicates the conditional probability of • given ♦ . Under the condition of local nonchaoticity of SNEE, the particle-gate interaction events are relatively independent of each other. The crossing ratio of the gate can be calculated as where {•} represents the probability of microstate •, and indicates the phase space. Notice that ⁼ { ψ _b }, and = {Φ _b }. Without extensive particle collision at the gate, there is no mechanism for the system to reach thermodynamic equilibrium. For example, for a freely-jointed rigid specular gate, in the forward process from ψ _a to ψ _b , upon the nonchaotic particle- gate interaction, v _b is determined by the following parameters: v _a , the length (L _G ) and the moment of inertia (I _G ) of the gate, the angular velocity ( ω _a ) and the swinging angle (Φ _a ) of the gate at ψ _a , the collision location on the gate (L _c ), the particle mass (m _P ) and the particle size (D _p ), and the incident angle of the particle ( ψ _a ). Thus, where represents a certain function. When the thermal motion of the gate is significant (i.e., ω _a ≠ 0), v _b is nonlinear to v _a , so that v _a and v _b do not have the same probability distribution; i.e., { ψ _b } { ψ _a which leads to { ψ _a }. For another example, if there is an attraction force (F _G ) between the gate and the “door stopper”, the gate would be self-closing. In other words, Φ _a tends to be a closed configuration. Notice that when D _P /L _G is nontrivial, Φ _b must be an open configuration, so is - With F _G , because the closed gate configuration is energetically favorable but the open configuration is energetically unfavorable, is unequal to {Φ _a }. Consequently, since the probability distributions of mismatch with those of ψ _a and Φ _a , in general, K ≠ 1.

The nonchaotic particle-gate interaction imposes a set of constraints on the probability of system microstates: , where p indicates probability, subscripts m and n indicate system microstates (m = 1,2,3 ... and n = m + 1, m + 2 ...), κ _mn = κ ^Nmn is the probability ratio, and N _mn is the excess number of particles on the gate side. At thermodynamic equilibrium, entropy S = — κ _B ∑ _i ρ _i lnρ _i reaches the maximum value (S _eq ) with two constraints on ρ _i , ∑ _i ρ _i = 1 and ∑ _i ρ _i E _i = U, where k _B is the Boltzmann constant, ∑ indicates summation for microstates (i = 1,2,3 ...), and are respectively the energy and the probability of the i-th microstate, and U is the internal energy. With the SNEE gate, the Lagrangian becomes are the Lagrange multipliers. To maximize = 0. Because of the constraints imposed by the SNEE gate, 5 reaches a local maximum, referred to as the maximum possible entropy of steady state (S _Q ). S _Q can be less than the maximum possible entropy of the system (5 _eq ).

Example 4: Spontaneous flow of particles

In Example 3, gas molecules spontaneously flow across the one-sidedly surface- grafted nanoporous membrane, leading to a flow of particles, until the steady-state is reached. In general, a nonequilibrium element can induce a continuous, semi- continuous, or cyclic “wind” (i.e., particle flow).

A Monte Carlo (MC) simulation was carried out. The program code was written and run in MATLAB® (MathWorks, Inc.).

As depicted by the diagrams in FIGs. 20A and 20B, the system under investigation is formed by a lower plain on the left-hand side, a vertical transition step, an upper plateau, a long (chaotic) ramp, and a lower plain on the right-hand side. A number of elastic particles randomly move in the system. There is no long-range force among the particles. The plateau height, i.e., the step size , is much less than the mean free path of the particles (λ _F ). The ramp size, , is much larger than λ _F . The in-plane direction from right to left is denoted by x; the in-plane direction vertical to x is denoted by the out-of-plane direction is z. The system is in a uniform gravitational field along -z, with the gravitational acceleration being denoted by g. The local dimension in the ramp from the plain to the plateau is denoted by z'.

The MC simulation system reflects the two-dimensional surfaces in FIGs. 20A- 20B and consisted of a rectangle box and a number of elastic particles, shown in FIG. 21. From left to right, it contains the left plain (“+”), the step, the plateau, the ramp, and the right plain The upper and the lower boundaries of the simulation box were rigid specular walls. The left and the right boundaries (AA’ and BB’) used periodic boundary condition. The width of the box (w ₀ ), i.e., the distance between the upper and the lower boundaries, was 500 Å. The direction from right to left was denoted by x. The length of plain “+” was 50 Å. The plateau length was 100 A. The transition step size, was 5 A. The ramp length, was 500 Å. The length of plain was 50 A.

The particles in the transition step were subjected to a gravitational force pointing to the left. The particles in the ramp were subjected to a gravitational force pointing to the right. The particle movement in the plain and on the plateau was not affected by the gravitational force. The particle-particle collision and the particle-wall collision were elastic. The main system parameters are listed below: the total particle number: N = 500; the initial system temperature on the plain: T = 1000 K; the particle mass: m = 1 g/mol = 1.66 × 10 ^-27 kg; the particle radius: r = 1 Å = 10 ^-10 m; the time step: Δt ₀ = 1 fs = 10 ^-15 s; the average particle kinetic energy on the plain: = κ _B T = 1.38 × 10 ^-20 J, where κ _B is the Boltzmann constant.

Initially, all the particles were randomly generated on the plain, with the initial velocities following the two-dimensional (2D) Maxwell-Boltzmann distribution. The moving direction is random. If the amplitude of the total initial x-dimension momentum of all the particles was larger than about 10 ^-3 p ₀ , the configuration would be rejected, where All the testing cases discussed below used the same initial condition. For each testing case, the effective gravitational acceleration in the transition step was and effective gravitational accelerations in the ramp was ,g _ramp ⁼ = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 1.0, 1.5, or 2.0. The Boltzmann factor is

Every time when a particle crossed the periodic boundary at the left-hand side of the simulation box (AA’), the timestep, the particle identification number (ID), the particle moving direction, and the particle velocity were recorded. For every 200 time steps, the position and the velocity of every particle were recorded.

The overall number of the particles crossing the left-hand side boundary (AA’) was calculated every 5000 time steps. The average flux from time step 2* 10 ⁴ to time step 1.5× 10 ⁵ is denoted by j. The error bars are calculated as the confidence interval, is the average value, n _t is the number of data points, s _t is the standard deviation, and δ _t is the inverse of Student’s t-distribution with the confidence level of 90%. The normalization factor of j was taken as where Δt is 5000 time steps, is the nominal average x-dimension particle velocity, and d ₀ = 705 Å is the total length of the simulation box. The result of j/j ₀ is given in FIG.

22.

For every 200 timesteps, the average x-dimension and γ-dimension particle momentums in the entire system were computed as where ∑ indicates summation of all the particles, and v _x and v _y are respectively the x- dimension and γ-dimension particle velocities. The result is shown in FIG. 23.

For every 5000 time steps, all the particles crossing the left-hand side boundary

(AA’) from right to left were used to calculate the effective partial pressure along the x direction, is the x-dimension velocity of a particle moving from right to left across AA’, and Δt is 5000 time steps. For every 5000 time steps, all the particles crossing the left-hand side boundary (AA’) from left to right were used to calculate the effective partial pressure along the -x direction, where v _x _ is the x-dimension velocity of a particle moving from left to right across AA’. Average P ₊ and P_ were calculated from time step 2× 10 ⁴ to time step 1.5 × 10 ⁵ . The net pressure across AA’ is defined as ΔP = P ₊ — P_. The numerical results are shown in FIG. 24, where the normalization factor is P ₀ = Nκ _B T /A _o .

Similar phenomenon can be realized by the charge carriers in a condensed matter. For instance, in metals, the mean free path of conduction electrons (λ _e ) is 20~50 nm; the Fermi velocity (v _F ) is on the scale of 10 ⁶ m/s, and the density of the conduction electrons (p _e ) is a few 10 ⁹ C/m ³ . Referring to FIG. 25, if a nanowire has an asymmetric structure with a small nano-step at one end and a wide slope at the other end, in an external electric field (E), a diffusive current can be spontaneously generated. The nano-step size should be much less than λ _e ; the slope size should be much larger than λ _e . Similar to j ₀ , the reference current density could be taken as j _e0 = ρ _e vF/2, on the scale of 10 ¹⁵ A/m ² . In accordance with Figure 1(C), the maximum current density might be a fraction of j _e0 , around 10 ¹⁴ A/m ² . If the cross-sectional area of a 1 μm-long nanowire is 10 nm ² , the current would be about 1 mA and the resistance is around 10 ³ ~10 ⁴ Ω. Thus, the nominal power density may be more than 10 ¹⁰ kW/cm ³ . A number of nano-steps can be placed in tandem or in parallel, to amplify the current or to reduce the requirement on E.

Example 5 : Spontaneous low-temperature to high-temperature heat transfer

A SNEE can be used to induce a spontaneous heat transfer from the cold side to the hot side of a system. One example is shown in FIG. 26A. A gas container is made of thermal insulating walls. The environment is an infinitely large reservoir of gas molecules and thermal energy, with constant temperature (T _c ) and gas pressure (P). There is a frictionless piston in the gas container, separating the container into two sections. The left section is connected to the environment either through a regular venting channel, or a molecular-sized one-sidedly-swinging gate (MOG). The MOG is similar to the one-sidedly surface-grafted membrane with bendable organic chains in Example 3. It can open to the inner side the container but cannot swing toward the outside. The right section is sealed, containing a high-temperature heat source that keeps the gas temperature in the right section at constant T _h ; T _h can be slightly higher than T _c . A low- temperature heat source keeps the gas temperature in the left section at constant T _c . The MOG is molecular-sized. The gas phase is an ideal gas. The MOG interacts with the gas molecules individually. Initially at state 1, the left section is open to the environment through the regular venting channel, and the MOG is blocked by a frictionless sliding cover. The gas pressure in the container is P. Then, the venting channel is closed by a frictionless sliding cover, and the MOG is exposed. Similar to Example 3, the gas pressure inside the gas container spontaneously increases. Denote the increased gas pressure in the container by P ⁺ . The right section is compressed by the piston. The piston does work to the sealed gas, by absorbing heat (Q ₁ ) from the low-temperature heat source. The same amount of heat is released to the high-temperature heat source. The released heat (Q ₁ ) equals to the piston work, W ₁ . After steady-state 2 is reached, the piston is then affixed by a frictionless locking pin. The MOG is covered, and the venting channel is open. The gas pressure in the left section is reduced back to P (state 3). Then, the locking pin is removed, allowing the piston to expand irreversibly, until the gas pressure in the right section is lowered to P. During the irreversible expansion process, the piston does work (W ₂ ) to the environment. Heat is absorbed from the high- temperature heat source (Q ₂ ), and the same amount of heat is releases heat to the low- temperature heat source; Q ₂ = W ₂ . Because W ₂ < W _1, Q ₂ < Q ₁ . Thus, in an operation cycle, there is an overall heat transfer, ΔQ = Q ₁ — Q ₂ , from the low-temperature heat source to the high-temperature heat source.

FIG. 26B illustrates another example of spontaneous heat transfer. A tubular channel forms a closed loop, which contains an idea gas. The tube wall is thermal insulating. Inside the tube, a low-temperature heat source maintains the internal temperature at T _c . Outside the tube, a high-temperature heat sink maintains the external temperature at T _h . T _h is higher than T _c . A MOG is installed in the tube. Across the MOG, gas spontaneously flows from the non-gated back side to the gated front side, leading to a circular gas flow. The gas flow activates a windmill that is connected to a viscous damper outside the tube. The connection between the inner windmill and the external damper is thermal insulating. The gas flow does work to the windmill, by absorbing heat (Q) from the low-temperature heat source. The work is dissipated by the damper, releasing heat (Q) to the high-temperature heat sink. Hence, heat is continuously transferred from the low-temperature end of the system (T _c ) to the high-temperature end (T _h ).

FIG. 27 depicts an example that is somewhat similar to FIG. 26B, but not based on the MOG. In a gravitational field (g), an asymmetric non-planar gap is immersed in an ideal gas. The gap is formed by two rigid non-planar surfaces. The gap has a nonchaotic step, with the step height much less than the mean free path of the gas molecules (λ _F ). The gap thickness is much less than z. The step connects a lower plain and a higher plateau. The right-hand side of the gap is a plain at the same level as the left-hand side. It is connected to the plateau through a wide ramp. The ramp width is much larger than λ _F . Thus, the gas molecular movement along the narrow step is nonchaotic, and the gas molecular movement along the ramp is chaotic and ergodic. Similar to the mechanism of Example 1, because the crossing ratios of gas molecules across the nonchaotic step and the chaotic ramp are different, a spontaneous gas flow would be generated from the left-hand side to the right-hand side. The gas flow could activate a windmill. The work that the gas flow does to the windmill (W) is from heat absorption from the environment (Q). The environment is an infinitely large reservoir of gas molecules and thermal energy, at constant temperature T. The windmill is connected to a viscous damper, and the work W is continuously dissipated. The damper is thermally insulated from the environment by a sealing box. Inside the box, a high-temperature heat sink keeps the inner temperature at constant T _h . As the damper dissipates energy, heat is released to the heat sink. Overall, heat continuously transfers from the low-temperature heat source (the environment) to the high-temperature heat source.

FIG. 28 depicts another example within a gravitational field (g) in which a non- planar gap contains ideal gas. The left-hand side of the gap is a lower plain. The right- hand side of the gap is a higher plateau. The plateau and the plain are connected through a narrow step. The step height is much less than the mean free path of the gas molecules (λ _F ), so that the gas molecular motion in the step is nonchaotic. The upper and the lower surfaces of the gap are rigid thermal insulating walls. The gap thickness is much less than the step height, . The gas temperature is kept at T _h by a heat reservoir in the plain, and at T _c by a different heat reservoir in the plateau. The plain size and the plateau size are much larger than λ _F . In the plain and the plateau, the gas molecular behavior is ergodic and chaotic. T _h ^can be slightly higher or lower than T _c . The nonchaotic step forms a SNEE. FIG. 28 shows spontaneous low-temperature to high- temperature heat transfer across a nonchaotic step when is relatively small. When is relatively large, the positions of the heat source and the heat sink should be shifted.

The difference between T _h and T _c is sufficiently small.

In the ideal-case scenario, the collision of the particles (i.e., the gas molecules) in the narrow step is negligible. Denote the vertical dimension by z, with the positive direction against the gravitational force. Assume that the particle velocity follows the Maxwell-Boltzmann distribution in the plain and the plateau. At the lower boundary of the transition step, the average z-dimension kinetic energy of the particles that can overcome the step is: where m is the particle mass, v _z indicates the z-dimension particle velocity, and κ _B is the Boltzmann constant. At the upper boundary of the transition step, the average z-dimension kinetic energy of the particles climbing from the plain is: The kinetic energy in the other two dimensions of these particles is the same as in the plain, i.e., κ _B T _c /2. It can be seen that , and increases with that is, the particles arriving at the plateau from the plain tend to have a higher effective temperature than T _c . The effective temperature increase may be assessed as If the difference between T _h and T _c ( ΔT) is less than ΔT _G , heat would be released to the heat reservoir on the plateau. The particles moving from the plateau to the plain would gain a kinetic energy of corresponding to an effective temperature high than Hence, these particles tend to release heat to the heat reservoir on the plain.

When and the overall heat transfer is from the low- temperature heat reservoir in the plain to the high-temperature heat reservoir on the plateau. If T _h > T _c while their difference is less than the difference between ΔT _P and ΔT _G , spontaneous cold-to-hot heat transfer would take place. When and the overall heat transfer is from the plateau to the plain. If T _h < T _c while the difference between them is less than the difference between ΔT _P and ΔT _G , spontaneous cold-to-hot heat transfer would occur.

In all the systems discussed above, the particles can be neutral, magnetic, or charged. The particles can be charge carriers, such as ions, electrons, holes, or any combination thereof. The particles can carry a magnetic field. The particles can be subatomic particles or fundamental particles. The particle can be a single atom or molecule, or contain multiple atoms or molecules. The particle can be zero-dimensional, one-dimensional, two-dimensional, or three-dimensional. When the system is isolated, its entropy cannot evolve away from the maximum possible steady-state value (S _Q ); that is, entropy would remain constant or converge toward S _Q .